On the Curves of Trisection. 345 
commended, and whether the Curves of Trisection do not 
offer a more simple and excellent method of trisecting an 
angle, than any method previously known, inasmuch as by 
means of a single curve every angle is trisected, and thus it 
is no longer necessary to describe a new curve for every dif- 
ferent angle: 
By means of the Quadratrix an angle may be trisected ; 
but the Quadratrix cannot be described by a continued mo- 
tion; and as, in order to describe it, the quadrant must be 
divided into equal parts ; it can be of no use in trisecting an 
angle, unless the angle in fact be previously trisected in 
forming the curve. The thing must be done before the 
Quadratrix can furnish-any aid in doing it. 
By the Trochoid or Cycloid, a curve which was not 
known by the ancients, angles may also be trisected ; but 
this curve, described by the motion of a wheel on a plane, 
is not easy and simple of description, and is of little practical 
use for the trisection of an angle. 
The Curves of 'Trisection may be distinguished by calling 
the first the Curve of Secants, and the second the Curve of 
Sines, since the first gives the Secant of the arc measuring 
the third of the angle to be trisected, and the second gives 
the point in the radius, from which point the Sine of the 
third of the proposed angle is to be drawn by a perpendicu- 
lar to the radius. 
1. The Trisecting Curve of Secants. 
In figure 1, the line FmoD is the Trisecting Curve of 
Secants, passing from F at the extremity of the radius CF 
to D at the extremity of the radius CD, which is double the 
radius CF or CE; and so passing, as that the intersection, 
at any point of this curve, of a straight line from the centre 
to the circumference (as CA) and ofa straight line from the 
extremity of the radius D to the circumference, (as DG) 
shall give the distance from the curve to the centre, (as oC) 
equal to the distance from the curve to the circumference, 
(as 0G.) 
1. This curve may be conceived to be generated, some- 
what like the quadratrix and spiral, in the following manner. 
If on the centre D the radius DB be supposed to revolve 
from B to G and g, and at the same time a point in this ra- 
